from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,2,7]))
pari: [g,chi] = znchar(Mod(125,1911))
Basic properties
| Modulus: | \(1911\) | |
| Conductor: | \(1911\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1911.cv
\(\chi_{1911}(83,\cdot)\) \(\chi_{1911}(125,\cdot)\) \(\chi_{1911}(356,\cdot)\) \(\chi_{1911}(398,\cdot)\) \(\chi_{1911}(629,\cdot)\) \(\chi_{1911}(671,\cdot)\) \(\chi_{1911}(902,\cdot)\) \(\chi_{1911}(944,\cdot)\) \(\chi_{1911}(1217,\cdot)\) \(\chi_{1911}(1448,\cdot)\) \(\chi_{1911}(1490,\cdot)\) \(\chi_{1911}(1721,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((638,1522,1471)\) → \((-1,e\left(\frac{1}{14}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 1911 }(125, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-i\) | \(e\left(\frac{1}{28}\right)\) |
sage: chi.jacobi_sum(n)